Given the analog part fixed, we proof the result only for \(\mathbf {V}_{i_l}\) . The result for \(\mathbf {V}_{i_r}\)  is then straightforward based on the proof for \(\mathbf {V}_{i_l}\) .
The proof relies on simplifying (REF ) with fixed analog part until the
Hadamard’s inequality applies. The Cholesky
decomposition of the matrix \(\mathbf {G}_{i_l}^H \big ( \hat{\mathbf {A}}_{i_l} + \hat{\mathbf {B}}_{i_l} + \lambda _{i_l} \mathbf {I}\big )\mathbf {G}_{i_l})\)  is written as
\(\mathbf {L}_{i_l} \mathbf {L}_{i_l}^H\)  where \(\mathbf {L}_{i_l}\)  is a lower-triangular Cholesky factor.
Upon defining \(\tilde{\mathbf {V}_{i_l}} = \mathbf {L}_{i_l}^H \mathbf {V}_{i_l}\) .A similar reasoning follows also for \(\mathbf {G}_{i_l}^H \mathbf {H}_{i_r,i_l}^H \mathbf {F}_{i_r}^H \mathbf {R}_{\overline{i_r}}^{-1}\mathbf {F}_{i_r} \mathbf {H}_{i_r,i_l} \mathbf {G}_{i_l} \) . By using the result provided in Proposition 1 [1]}, with the matrices \(\mathbf {G}_{i_l}^H \big ( \hat{\mathbf {A}}_{i_l} + \hat{\mathbf {B}}_{i_l} + \lambda _{i_l} \mathbf {I}\big )\mathbf {G}_{i_l})\)  and \(\mathbf {G}_{i_l}^H \mathbf {H}_{i_r,i_l}^H \mathbf {F}_{i_r}^H \mathbf {R}_{\overline{i_r}}^{-1}\mathbf {F}_{i_r} \mathbf {H}_{i_r,i_l} \mathbf {G}_{i_l} \) , it follows immediately that the optimal digital beamformer \(\mathbf {V}_{i_l}\)  are given as a dominant generalized eigenvectors of these two matrices. The same reasoning follows also for the digital beamformer \(\mathbf {V}_{i_r}\) .

Note the GEV solution provides the optimal beamforming directions but not the optimal power allocation. Therefore, to include the optimal power allocation, we normalize the columns of the digital beamformers to be unit norm.
